20 November 2007

the Tao of compactness

Terence Tao on compactness and compactification (from the forthcoming Princeton Companion to Mathematics). Particularly interesting is the fact that using ideas of compactification, one can rigorously view a continuous object as a limit of discrete objects: the circle group as a limit of cyclic groups, straight lines as the limit of circles (they're just circles with the center at infinity!), the Dirac delta function as the limit of taller and narrower "spike functions", and so on.

Since I seem to be on a nomenclature kick, "compact" is kind of misleading nomenclature at first; if you didn't know what "compact" meant, wouldn't you think that if the closed interval [0, 1] is compact then the open interval (0, 1) surely must be? After all, "compact" means something like "small" in everyday discourse, and (0, 1) is contained in [0, 1]. It seems even sillier when we say that the real line is compactified by adding a "point at infinity"... until you realize that this means you can just join the edges together and turn the line into a circle, and circles are "smaller" than lines. Intuitively "compact" should mean what "bounded" means, and of course the Heine-Borel theorem says that in Euclidean space "compact" means "closed and bounded". Why should something called "compact" and something called "closed" be connected? And why aren't "closed" and "open" logically opposite? When I first learned the definitions of those -- the definitions on the real line, in a real analysis class -- I must have flipped back to the page with those definitions on them in baby Rudin at least once a day. I eventually made my peace with those definitions... but it took a while.